Affine insertion and Pieri rules for the affine Grassmannian

Abstract

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Gr"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm {Gr}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis n comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL(n,\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our main results are: Pieri rules for the Schubert bases of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis Gr right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^*(\textrm {Gr})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript asterisk Baseline left-parenthesis Gr right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_*(\textrm {Gr})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Schur functions, which represent the Schubert basis of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript asterisk Baseline left-parenthesis Gr right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_*(\textrm {Gr})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A combinatorial interpretation of the pairing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis Gr right-parenthesis times upper H Subscript asterisk Baseline left-parenthesis Gr right-parenthesis right-arrow double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo> × </mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">H^*(\textrm {Gr})\times H_*(\textrm {Gr}) \rightarrow \mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induced by the cap product. These results are obtained by interpreting the Schubert bases of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Gr"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>Gr</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textrm {Gr}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously. Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.